3.2709 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx\)

Optimal. Leaf size=222 \[ \frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{567 (3 x+2)^{7/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac{1864 \sqrt{1-2 x} (5 x+3)^{3/2}}{6615 (3 x+2)^{5/2}}+\frac{17830424 \sqrt{1-2 x} \sqrt{5 x+3}}{8751645 \sqrt{3 x+2}}-\frac{558524 \sqrt{1-2 x} \sqrt{5 x+3}}{1250235 (3 x+2)^{3/2}}-\frac{1717916 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8751645}-\frac{17830424 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8751645} \]

[Out]

(-558524*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1250235*(2 + 3*x)^(3/2)) + (17830424*Sqrt
[1 - 2*x]*Sqrt[3 + 5*x])/(8751645*Sqrt[2 + 3*x]) - (1864*Sqrt[1 - 2*x]*(3 + 5*x)
^(3/2))/(6615*(2 + 3*x)^(5/2)) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(27*(2 + 3*
x)^(9/2)) + (362*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(567*(2 + 3*x)^(7/2)) - (1783042
4*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/8751645 - (17179
16*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/8751645

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Rubi [A]  time = 0.505333, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{567 (3 x+2)^{7/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac{1864 \sqrt{1-2 x} (5 x+3)^{3/2}}{6615 (3 x+2)^{5/2}}+\frac{17830424 \sqrt{1-2 x} \sqrt{5 x+3}}{8751645 \sqrt{3 x+2}}-\frac{558524 \sqrt{1-2 x} \sqrt{5 x+3}}{1250235 (3 x+2)^{3/2}}-\frac{1717916 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8751645}-\frac{17830424 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8751645} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(11/2),x]

[Out]

(-558524*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1250235*(2 + 3*x)^(3/2)) + (17830424*Sqrt
[1 - 2*x]*Sqrt[3 + 5*x])/(8751645*Sqrt[2 + 3*x]) - (1864*Sqrt[1 - 2*x]*(3 + 5*x)
^(3/2))/(6615*(2 + 3*x)^(5/2)) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(27*(2 + 3*
x)^(9/2)) + (362*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(567*(2 + 3*x)^(7/2)) - (1783042
4*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/8751645 - (17179
16*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/8751645

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Rubi in Sympy [A]  time = 47.8188, size = 201, normalized size = 0.91 \[ - \frac{14318 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{138915 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{362 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3969 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{27 \left (3 x + 2\right )^{\frac{9}{2}}} + \frac{17830424 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{8751645 \sqrt{3 x + 2}} + \frac{858958 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1250235 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{17830424 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{26254935} - \frac{18897076 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{306307575} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**(11/2),x)

[Out]

-14318*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(138915*(3*x + 2)**(5/2)) - 362*(-2*x + 1
)**(3/2)*(5*x + 3)**(3/2)/(3969*(3*x + 2)**(7/2)) - 2*(-2*x + 1)**(3/2)*(5*x + 3
)**(5/2)/(27*(3*x + 2)**(9/2)) + 17830424*sqrt(-2*x + 1)*sqrt(5*x + 3)/(8751645*
sqrt(3*x + 2)) + 858958*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1250235*(3*x + 2)**(3/2))
- 17830424*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/26254935
- 18897076*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/30630757
5

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Mathematica [A]  time = 0.431317, size = 110, normalized size = 0.5 \[ \frac{\frac{24 \sqrt{1-2 x} \sqrt{5 x+3} \left (722132172 x^4+2043155529 x^3+2115318249 x^2+955601637 x+159578303\right )}{(3 x+2)^{9/2}}+8 \sqrt{2} \left (5257595 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+8915212 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{105019740} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(11/2),x]

[Out]

((24*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(159578303 + 955601637*x + 2115318249*x^2 + 204
3155529*x^3 + 722132172*x^4))/(2 + 3*x)^(9/2) + 8*Sqrt[2]*(8915212*EllipticE[Arc
Sin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 5257595*EllipticF[ArcSin[Sqrt[2/11]*Sqrt
[3 + 5*x]], -33/2]))/105019740

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Maple [C]  time = 0.033, size = 624, normalized size = 2.8 \[ -{\frac{2}{262549350\,{x}^{2}[phone]\,x-78764805} \left ( 425865195\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-21663965160\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -63461062386\,{x}^{5}-63089824509\,{x}^{4}-16625604096\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+1436204727 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(11/2),x)

[Out]

-2/26254935*(425865195*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2
*I*11^(1/2)*3^(1/2)*2^(1/2))*x^4*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+72213
2172*2^(1/2)*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2
)*2^(1/2))*x^4*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*Elli
pticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x^3*(1
-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)[phone]*2^(1/2)*EllipticE(1/11*11^(1/
2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x^3*(1-2*x)^(1/2)*(3+5*
x)^(1/2)*(2+3*x)^(1/2)[phone]*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x
)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^
(1/2)[phone]*2^(1/2)*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^
(1/2)*3^(1/2)*2^(1/2))*x^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*2
^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1
/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*EllipticE(1/1
1*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x*(3+5*x)^(1/2)
*(2+3*x)^(1/2)*(1-2*x)^(1/2)-21663965160*x^6+84121520*2^(1/2)*(3+5*x)^(1/2)*(2+3
*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(
1/2)*3^(1/2)*2^(1/2))[phone]*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2
)*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))-
63461062386*x^5-63089824509*x^4-16625604096*x^3+11383710240*x^2+8121679824*x+143
6204727)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(9/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(11/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(11/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(11/2),x, algorithm="fricas")

[Out]

integral(-(50*x^3 + 35*x^2 - 12*x - 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((243*x^5 +
810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*sqrt(3*x + 2)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(11/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(11/2), x)